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Mirrors > Home > MPE Home > Th. List > Mathboxes > ehl2eudisval0 | Structured version Visualization version GIF version |
Description: The Euclidean distance of a point to the origin in a real Euclidean space of dimension 2. (Contributed by AV, 26-Feb-2023.) |
Ref | Expression |
---|---|
ehl2eudisval0.e | ⊢ 𝐸 = (𝔼hil‘2) |
ehl2eudisval0.x | ⊢ 𝑋 = (ℝ ↑m {1, 2}) |
ehl2eudisval0.d | ⊢ 𝐷 = (dist‘𝐸) |
ehl2eudisval0.0 | ⊢ 0 = ({1, 2} × {0}) |
Ref | Expression |
---|---|
ehl2eudisval0 | ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5333 | . . . 4 ⊢ {1, 2} ∈ V | |
2 | ehl2eudisval0.0 | . . . . 5 ⊢ 0 = ({1, 2} × {0}) | |
3 | ehl2eudisval0.x | . . . . 5 ⊢ 𝑋 = (ℝ ↑m {1, 2}) | |
4 | 2, 3 | rrx0el 24001 | . . . 4 ⊢ ({1, 2} ∈ V → 0 ∈ 𝑋) |
5 | 1, 4 | mp1i 13 | . . 3 ⊢ (𝐹 ∈ 𝑋 → 0 ∈ 𝑋) |
6 | ehl2eudisval0.e | . . . 4 ⊢ 𝐸 = (𝔼hil‘2) | |
7 | ehl2eudisval0.d | . . . 4 ⊢ 𝐷 = (dist‘𝐸) | |
8 | 6, 3, 7 | ehl2eudisval 24026 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝐹𝐷 0 ) = (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)))) |
9 | 5, 8 | mpdan 685 | . 2 ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)))) |
10 | 1ex 10637 | . . . . . . . . . . . 12 ⊢ 1 ∈ V | |
11 | 2ex 11715 | . . . . . . . . . . . 12 ⊢ 2 ∈ V | |
12 | c0ex 10635 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
13 | xpprsng 6902 | . . . . . . . . . . . 12 ⊢ ((1 ∈ V ∧ 2 ∈ V ∧ 0 ∈ V) → ({1, 2} × {0}) = {〈1, 0〉, 〈2, 0〉}) | |
14 | 10, 11, 12, 13 | mp3an 1457 | . . . . . . . . . . 11 ⊢ ({1, 2} × {0}) = {〈1, 0〉, 〈2, 0〉} |
15 | 2, 14 | eqtri 2844 | . . . . . . . . . 10 ⊢ 0 = {〈1, 0〉, 〈2, 0〉} |
16 | 15 | fveq1i 6671 | . . . . . . . . 9 ⊢ ( 0 ‘1) = ({〈1, 0〉, 〈2, 0〉}‘1) |
17 | 1ne2 11846 | . . . . . . . . . 10 ⊢ 1 ≠ 2 | |
18 | 10, 12 | fvpr1 6952 | . . . . . . . . . 10 ⊢ (1 ≠ 2 → ({〈1, 0〉, 〈2, 0〉}‘1) = 0) |
19 | 17, 18 | ax-mp 5 | . . . . . . . . 9 ⊢ ({〈1, 0〉, 〈2, 0〉}‘1) = 0 |
20 | 16, 19 | eqtri 2844 | . . . . . . . 8 ⊢ ( 0 ‘1) = 0 |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → ( 0 ‘1) = 0) |
22 | 21 | oveq2d 7172 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − ( 0 ‘1)) = ((𝐹‘1) − 0)) |
23 | eqid 2821 | . . . . . . . . 9 ⊢ {1, 2} = {1, 2} | |
24 | 23, 3 | rrx2pxel 44747 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘1) ∈ ℝ) |
25 | 24 | recnd 10669 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘1) ∈ ℂ) |
26 | 25 | subid1d 10986 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − 0) = (𝐹‘1)) |
27 | 22, 26 | eqtrd 2856 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − ( 0 ‘1)) = (𝐹‘1)) |
28 | 27 | oveq1d 7171 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → (((𝐹‘1) − ( 0 ‘1))↑2) = ((𝐹‘1)↑2)) |
29 | 15 | fveq1i 6671 | . . . . . . . 8 ⊢ ( 0 ‘2) = ({〈1, 0〉, 〈2, 0〉}‘2) |
30 | 11, 12 | fvpr2 6953 | . . . . . . . . 9 ⊢ (1 ≠ 2 → ({〈1, 0〉, 〈2, 0〉}‘2) = 0) |
31 | 17, 30 | mp1i 13 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝑋 → ({〈1, 0〉, 〈2, 0〉}‘2) = 0) |
32 | 29, 31 | syl5eq 2868 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → ( 0 ‘2) = 0) |
33 | 32 | oveq2d 7172 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − ( 0 ‘2)) = ((𝐹‘2) − 0)) |
34 | 23, 3 | rrx2pyel 44748 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘2) ∈ ℝ) |
35 | 34 | recnd 10669 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘2) ∈ ℂ) |
36 | 35 | subid1d 10986 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − 0) = (𝐹‘2)) |
37 | 33, 36 | eqtrd 2856 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − ( 0 ‘2)) = (𝐹‘2)) |
38 | 37 | oveq1d 7171 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → (((𝐹‘2) − ( 0 ‘2))↑2) = ((𝐹‘2)↑2)) |
39 | 28, 38 | oveq12d 7174 | . . 3 ⊢ (𝐹 ∈ 𝑋 → ((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)) = (((𝐹‘1)↑2) + ((𝐹‘2)↑2))) |
40 | 39 | fveq2d 6674 | . 2 ⊢ (𝐹 ∈ 𝑋 → (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2))) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
41 | 9, 40 | eqtrd 2856 | 1 ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 {csn 4567 {cpr 4569 〈cop 4573 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 − cmin 10870 2c2 11693 ↑cexp 13430 √csqrt 14592 distcds 16574 𝔼hilcehl 23987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-rnghom 19467 df-drng 19504 df-field 19505 df-subrg 19533 df-staf 19616 df-srng 19617 df-lmod 19636 df-lss 19704 df-sra 19944 df-rgmod 19945 df-cnfld 20546 df-refld 20749 df-dsmm 20876 df-frlm 20891 df-nm 23192 df-tng 23194 df-tcph 23773 df-rrx 23988 df-ehl 23989 |
This theorem is referenced by: ehl2eudis0lt 44762 itscnhlinecirc02plem3 44820 |
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